Another option if you’re worried about traffic would be drive to Papakura and take the train into the hospital. Trains have been pretty reliable since they went electric but obviously nothing is perfect.

Train would require a transfer at Newmarket to the western line but that shouldn’t be too difficult.

It is school holidays, I'm unsure of what difference that makes on the Southern Motorway in the morning (it has made the Northern heading into the city a nightmare after work).

How long does the trip normally take assuming traffic is flowing freely? In the morning it can take anywhere from 40 minutes to an hour to get from Drury to Mt Wellington, then about another 20-30 minutes from there to Khyber Pass. I'd say adding an hour to your normal travel time will get you there with plenty to spare. Myself, I prefer to err on the side of arriving too early than stressing about arriving late.

Yep that 4 hours is pretty accurate. As others have said, add an hour for potential motorway issues depending on how important the appointment is, and half an hour for at least one rest stop (possibly Ngatea or Bombays) and your 3am leave would be perfect.

Only found out yesterday about a gap at 8.30 tomorrow (for the appointment), so imagine last minute flying would be a bit expensive, same I imagine for last minute hotel room(s). Maybe if we had (quite a) bit more warning. Such is life.